Admissible Bayes equivariant estimation of location vectors for spherically symmetric distributions with unknown scale
Abstract
This paper investigates estimation of the mean vector under invariant quadratic loss for a spherically symmetric location family with a residual vector with density of the form , where is unknown. We show that the natural estimator is admissible for . Also, for , we find classes of generalized Bayes estimators that are admissible within the class of equivariant estimators of the form . In the Gaussian case, a variant of the James–Stein estimator, , which dominates the natural estimator , is also admissible within this class. We also study the related regression model.
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t1This work was partially supported by KAKENHI #25330035, #16K00040. t2This work was partially supported by grants from the Simons Foundation (#209035 and #418098 to William Strawderman).
1 Introduction
Let
| (1.1) |
where and and where and are unknown. We mainly assume
| (1.2) |
and consider the problem of estimating under scaled quadratic loss
| (1.3) |
In particular, we are interested in the admissibility among the class of equivariant estimators of the form
| (1.4) |
We assume that is defined so that each coordinate has variance . In particular, this implies that in (1.1), satisfies
| (1.5) |
for . Needless to say, this is a generalization of the Gaussian case where
and hence and are mutually independent.
In Section 2, we show that if an estimator of the form
| (1.6) |
is admissible within the class of all such estimators, then it is also admissible within a larger class of estimators, the class of all estimators of the form given by (1.6). Note that the risk of an equivariant estimator of the form (1.6) is a function of . Section 3 studies equivariant estimators of the form (1.6) which minimize the average risk with respect to a (proper) prior on the maximal invariant . We give an expression for this average risk and for the equivariant estimator which effects the minimization. Additionally we show that this proper Bayes equivariant estimator is equivalent to the generalized (and not proper) Bayes estimator corresponding to the prior on
Further we demonstrate that such an estimator is admissible among the class of estimators of the form (1.6) and hence (1.4).
Section 4, using Blyth’s (1951) method, extends the class of estimators which are admissible within the class of estimators of the form (1.6). One main result gives admissibility under including
for densities including the normal distribution and many generalized multivariate distributions. An interesting special case gives admissibility (within the class of equivariant estimators) of the generalized Bayes equivariant estimator corresponding to or . Here the form of the generalized Bayes estimator is independent of the underlying density , as shown in Maruyama (2003). Further this estimator is minimax and dominates the James and Stein (1961) estimator
provided is non-increasing. Another interesting result is on a variant of the James–Stein estimator of the simple form
In the Gaussian case, this is the generalized Bayes equivariant estimator corresponding to
It is admissible within the class of equivariant estimators, and is minimax.
In Section 5, we demonstrate that our setting (1.1) is regarded as a canonical form of a regression model with an intercept and a general spherically symmetric error distribution, where estimators of the form (1.6) corresponds to estimators of the vector of regression coefficients of the form where , is the vector of least square estimators, and is the coefficient of determination. Also estimators of the form (1.4) corresponds to estimators of the vector of regression coefficients of the form where and is the vector of values. Hence, from the regression viewpoint, these two classes of equivariant estimators are quite natural.
Section 6 gives some concluding remarks. Most of the proofs are given in a series of Appendices.
In this paper, we consider admissibility within the (restricted) class of equivariant estimators. The ultimate goal in this direction is clearly to show that some equivariant estimators of the form (1.6) are admissible among all estimators. Actually, when , the natural estimator is admissible among all estimators, as shown in Appendix M. (This is not surprising but very expected. As far as we know, however, admissibility of with nuisance unknown scale , has not yet reported in any major journals and hence we provide the proof.) When the Stein phenomenon with occurs, considering general admissibility of equivariant estimators in the presence of nuisance parameter(s) has been a longstanding unsolved problem as mentioned in James and Stein (1961) and Brewster and Zidek (1974). While our results with do not resolve the general admissibility issue, they do advance substantially our understanding of admissibility within the class of equivariant estimators.
2 Admissibility in a broader sense
We consider two groups of transformations. In the following, let .
- Group I
-
where , the group of orthogonal matrices, and .
- Group II
-
where .
Equivariant estimators for Group I should satisfy
and reduce to estimators of the form
| (2.1) |
where . Equivariant estimator for Group II should satisfy
and reduce to estimator of the form
| (2.2) |
where . It is useful to note the following.
Lemma 2.1.
-
1.
The risk, , of an estimator , is a function of .
-
2.
The risk, , of an estimator , is a function of .
The standard proof is left to the reader.
Let two classes of estimators be
Clearly it follows that . We shall show that if is admissible among the class , then it is admissible among the class . The proof is due to Section 3 of Stein (1956), based on the compactness of the orthogonal group , and the continuity of, the problem.
Theorem 2.1.
If is admissible among the class , then it is admissible among the class .
Proof.
See Appendix A. ∎
In this paper, we will investigate admissibility among the class . Admissibility admissibility among the class then follows by Theorem 2.1.
3 Proper Bayes equivariant estimators
Recall that an equivariant estimator for Group I is given by
| (3.1) |
Since, as noted in Lemma 2.1, the risk function of the estimator , , depends only on , it may be expressed as
| (3.2) |
Let . We assume the prior density on is , and in this section, we assume the propriety of , that is,
| (3.3) |
For an equivariant estimator , we define the Bayes equivariant risk as
| (3.4) |
In this paper, the estimator which minimizes , is called the Bayes equivariant estimator and is denoted by . In the following, let
| (3.5) |
and
| (3.6) |
so that is a proper probability density on , that is,
| (3.7) |
Theorem 3.1.
Assume and that satisfies (1.5).
- 1.
-
2.
Given , the minimizer of with respect to is
(3.10) -
3.
The Bayes equivariant estimator
(3.11) is equivalent to the generalized Bayes estimator of with respect to the joint prior density where .
-
4.
The Bayes equivariant estimator is admissible among the class .
Proof.
See Appendix B. ∎
Remark 3.1.
As shown in Appendix C, the generalized Bayes estimator of with respect to the joint prior density for any is a member of the class . Part 3 of Theorem 3.1 applies only to the special case of . The admissibility results of this section and of Section 4 apply only to this special case of and imply neither admissibility or inadmissibility of generalized Bayes estimators if . Also note that while is assumed proper in this section, the prior on , , is never proper since
4 Admissible Bayes equivariant estimators through the Blyth method
Even if on (and hence on ) is improper, that is
| (4.1) |
the estimator given by (3.11) can be defined if and given by (3.9) are both finite, and the admissibility of within the class of equivariant estimators can be investigated through the Blyth method.
We consider the Bayes equivariant risk difference under which is proper, but not necessarily standardized; i.e., . Let and be Bayes equivariant estimators with respect to and , respectively. By Parts 1 and 2 of Theorem 3.1, the Bayes equivariant risk difference under is given as follows:
| (4.2) | |||
where is given by (3.5) and where
| (4.3) |
There are several versions of the Blyth method. For our purpose, the following version from Brown (1971) and Brown and Hwang (1982) is useful.
Theorem 4.1.
Assume that the sequence for satisfies
-
BL.1
for any and .
-
BL.2
for any fixed .
-
BL.3
for some positive .
-
BL.4
.
Then is admissible among the class .
Proof.
See Appendix D. ∎
We consider the following assumptions on in addition to (4.1).
Assumptions on .
-
A.1
is differentiable.
-
A.2
(Behavior around the origin) For , there exist and such that
where
-
A.3
(Asymptotic behavior) Let . Either A.3A.3.1 or A.3A.3.2 is assumed;
-
A.3.1
-
A.3.2
. Further either A.3(A.3.2)A.3.2.1 or A.3(A.3.2)A.3.2.2 is assumed;
-
A.3.2.1
is eventually monotone increasing and approaches from below.
-
A.3.2.2
.
-
A.3.2.1
-
A.3.1
A typical prior satisfying Assumptions A.1–A.3, corresponding to a generalized Strawderman’s (1971) prior, is given by
| (4.4) |
Assumptions A.1–A.3 are satisfied when or . See Appendix L for the proof. Note that the power prior for , which will be considered in Section 4.1, corresponds to the case and .
For a generalized prior satisfying Assumptions A.1–A.3, consider the sequence given by where , for and is defined by
| (4.5) |
and . It is clear that satisfies BL.1 of Theorem 4.1. In Lemma E.2 of Appendix E, we show that also satisfies BL.2 and BL.3 of Theorem 4.1.
For BL.4, note that given by (4.2) is a functional of as well as and . Some additional assumptions on (as well as (1.5)) are required as follows;
Assumptions on .
We note that, in addition to the normal distribution,
an interesting flatter tailed class, also satisfying Assumptions F.1–F.3, is given by the multivariate generalized Student with
For Assumptions F.3F.3.1 and F.3F.3.2, and are needed respectively.
The main result on admissibility of given by (3.11) among the class through the Blyth method is as follows.
Theorem 4.2.
- Case I
- Case II
The proof of Theorem 4.2, or essentially, equivalently the proof of BL.4,
under the above Assumptions, is provided in Appendices F, G and H. Prior to these sections, some preliminary needed results on , and are given in Appendix E.
Remark 4.1.
The basic idea behind the sequence given by (4.5) comes from the of Brown and Hwang (1982),
| (4.6) |
A smoothed version of the above is
| (4.7) |
The sequence given by (4.5) is more slowly changing in both and , in order to handle priors with flatter tail than treated in Brown and Hwang (1982). Also, with smooth , the proofs become simpler.
Remark 4.2.
Assumption A.3 is a sufficient condition for
| (4.8) |
which is related to admissibility in the known variance case as follows. Maruyama (2009) showed that, in the problem of estimating of , regularly varying priors with
| (4.9) |
lead to admissibility, that is, the (generalized) Bayes estimator
where
| (4.10) |
is admissible. As Maruyama (2009) pointed out, the sufficient condition (4.9), which depends directly on the prior , is closely related to Brown’s (1971) sufficient condition for admissibility
which depends on the marginal distribution and only indirectly on the prior. Note also that Assumption A.3 is tight for the non-integrability of (4.8), in the sense that, among the class with ,
where in the first expression satisfies Assumption A.3, and in the second, does not satisfy Assumption A.3. Actually, in the third case, , the corresponding Bayes equivariant estimator is inadmissible as shown in Maruyama and Strawderman (2017).
4.1 Some interesting cases
Here we present three interesting special cases of our main general theorem.
Corollary 4.1.
Assume Assumptions F.1, F.2 and F.3F.3.2 on .
-
1.
Then with , or equivalently the generalized Bayes estimator under the prior on given by
is admissible among the class .
-
2.
The form of the generalized Bayes estimator does not depend on and is given by where and
-
3.
This estimator is minimax simultaneously for all such .
-
4.
This estimator dominates the James–Stein estimator
if is nonincreasing.
Proof.
Corollary 4.2.
Proof.
The following corollary relates to the so-called “simple Bayes estimators” from Maruyama and Strawderman (2005).
Corollary 4.3.
Assume is Gaussian. Then the simple Bayes estimator
with and is admissible among the class . Furthermore, the estimator with is minimax.
5 Canonical form of the regression setup
Suppose a linear regression model is used to relate to the predictors ,
| (5.1) |
where is an unknown intercept parameter, is an vector of ones, is an design matrix, and is a vector of unknown regression coefficients. In the error term, is an unknown scalar and has a spherically symmetric distribution,
| (5.2) |
where is the probability density, , and . Hence the density of is
| (5.3) |
where satisfies
for . We assume that the columns of have been centered so that for . We also assume that and are linearly independent, which implies that
Let be an orthogonal matrix of the form
where is matrix which satisfies , and . Also let where .
Let
where . Then are sufficient and the joint density of is
where . Further the marginal density of is
which we are considering in this paper, where and
Note that the loss function corresponds to so-called “predictive loss” for estimation of the regression coefficient vector .
In the equivariant estimator of
is in the regression context where is the coefficient of determination. It is natural to make use of for shrinkage since small corresponds to less reliability of the least squares estimator of . We note that the corresponding “simple Bayes estimator” for regression coefficient is rewritten as
and has a shrinkage factor which is increasing in .
In the equivariant estimator ,
| (5.4) |
where and is a vector of the -values.
6 Concluding remarks
We have established admissibility of certain generalized Bayes estimators within the class of equivariant estimators, of the mean vector for a spherically symmetric distribution with unknown scale under invariant loss. In some cases, we establish simultaneous minimaxity and, equivariant admissibility for broader classes of sampling distributions. In the Gaussian case we establish admissibility within the equivariant estimators of a class of generalized Bayes minimax estimators of a particularly simple form. We have also investigated similar issues in the setting of a general linear regression model with intercept and spherically symmetric error distribution. In this setting, the shrinkage factor of equivariant estimators of the regression coefficients depends on the coefficient of determination.
Appendix A Proof of Theorem 2.1
Suppose the estimator is strictly better than the estimator , that is,
| (A.1) |
for all with strict inequality for some value. Because of the continuity of and , strict inequality will hold for in some nonempty open set . The inequality (A.1) will remain true if is replaced by with orthogonal, since
Thus, for fixed , the set of for which
will be a nonempty open set. Let be the invariant probability measure on which assigns strictly positive measure to any nonempty open set (for the existence of such a measure, see Chapter 2 of Weil (1940)). Then the weighted estimator
is a member of the class , and because of the convexity of the loss function in , we have
with strict inequality for . This implies that is not admissible among as assumed and hence completes the proof.
Appendix B Proof of Theorem 3.1
[Parts 1 and 2] The Bayes equivariant risk given by (3.4) is rewritten as
| (B.1) |
where the third equality follows from (3.2). Further given by (B.1) is expanded as
| (B.2) |
Note that, by (1.5) and the propriety of the prior given by (3.7), the third term is equal to , that is,
| (B.3) |
The first and second terms of (B.2) with for respectively, are rewritten as
| (B.4) | |||
where is given by (3.5), , is the Jacobian, and
| (B.5) |
Similarly, the third term of (B.2) is rewritten as
| (B.6) | |||
where
| (B.7) |
Hence, by (B.3), (B.4) and (B.6), we have
| (B.8) |
Then the Bayes equivariant solution or minimizer of is
| (B.9) |
and hence the corresponding Bayes equivariant estimator is
| (B.10) |
where .
[Part 3] The generalized Bayes estimator of with respect to the density on ,
is given by
| (B.11) |
By change of variables and , we have
Comparing with given by (B.10), we see that with is
The second equality follows since is proportional to and the length of is .
[Part 4] Since the quadratic loss function is strictly convex, the Bayes solution is unique, and hence admissibility within follows.
Appendix C Proof that
As in (B.11), the generalized Bayes estimator of with respect to is given by
The estimator with and is
and, by change of variables and , is rewritten as
Hence .
Appendix D Proof of Theorem 4.1
Suppose that is inadmissible among the class and hence satisfies for all with strict inequality for some . Let . Clearly is also a member of . Then, using Jensen’s inequality, we have
for any . Since and are both continuous functions of , there exists an such that for . Then
which contradicts as .
Appendix E Preliminary results on , and
E.1 Preliminary results on
Lemma E.1.
Proof.
[Part 2] We have
| (E.1) |
[Part 3] As in (E.1) of Part 2, we have
[Part 4] By Assumption A.3A.3.1, there exist and such that
for all and hence we have
| (E.2) |
for , which implies that
| (E.3) |
Hence we have
| (E.4) |
[Part 7] Under Assumptions A.3A.3.1, by (E.3), as . Under Assumption A.3(A.3.2)A.3.2.1, it is clear that is bounded. Under Assumption A.3(A.3.2)A.3.2.2, there exist and such that
| (E.5) |
for all . As in (E.2) and (E.3), we have
and hence
| (E.6) |
which completes the proof.
[Part 8] Under Assumption A.3(A.3.2)A.3.2.1, there exists such that for is monotone decreasing. Then for is expressed by
and
Under Assumption A.3(A.3.2)A.3.2.2, by (E.6), we have
which completes the proof. ∎
E.2 The sequence
The function in (4.5) satisfies the following.
Lemma E.2.
Proof.
[Part 1] The part is straightforward given the form of .
[Part 4] At , is
which is greater than
[Part 5] The upper bound of , derived in Part 3, is decreasing in and hence
Further we have
and hence .
E.3 Assumption on
Lemma E.3.
Proof.
[Part 11.A] By (E.8), there exist and such that
| (E.11) |
for all . Then, by (E.11), we have
| (E.12) |
Further, by (E.11), we have
for all , and hence
| (E.13) |
By an integration by parts, the left-hand side is rewritten as
where the second equality follows from by (E.12). Then the inequality (E.13) is equivalent to
| (E.14) |
[Part 11.B] By Assumption F.1, we have
| (E.15) |
Also the integrability given by (1.5),
implies
| (E.16) |
| (E.17) |
Note by Assumption F.1. Also by (E.14) and (E.17), it follows that there exists such that
| (E.18) |
By (E.18), for , we have
| (E.19) |
and hence
| (E.20) |
| (E.21) |
for and hence
| (E.22) |
Combining (E.20) and (E.22), completes the proof of Part 11.B.
[Part 2] Note, by Part 1 of this lemma with ,
| (E.23) |
To prove Part 2, it suffices to show that, for ,
| (E.24) |
and also that there exist and such that
| (E.25) |
for all .
[Bound in (E.24)] Note is decomposed as
| (E.26) |
where is from (E.11). The first and third terms both are integrable since, by (E.19),
| (E.27) |
and by (E.21),
| (E.28) |
By (E.26), (E.27) and (E.28), we have . Then, by continuity of , it follows that
| (E.29) |
Further the integrability of follows since
| (E.30) |
Then, by (E.29) and (E.30), we have
Hence the bound in (E.24) is established.
[Bound in (E.25)] Let where is from (E.11). Under the decomposition of the integral region,
we have
For the region , implies and hence
which is bounded by (E.30). Similarly, for , since , we have
For the region
we have
Hence
| (E.31) |
where
| (E.32) |
Recall the assumption and hence note
| (E.33) |
By (E.21), the integrand of , for , is bounded as
and hence for is bounded as
| (E.34) |
where
Then, for any with ,
| (E.35) |
where the first and second inequalities follow from (E.34) and (E.33), respectively. By (E.31), (E.32), and (E.35),
which is bounded under the assumption . ∎
Appendix F Preliminary results for completing Proof of Theorem 4.2
Note that the first three parts BL.1, BL.2 and BL.3 of Blyth’s (1951) conditions needed to prove Theorem 4.2 follow from Parts 1, 8 and 6 of Lemma E.2, respectively. In this appendix we provide an alternative expression in BL.4. The proof of Theorem 4.2’s two cases, I and II is completed in the two succeeding sections G and H respectively using this re-expression.
| (F.1) |
with
The numerator of is rewritten as
| (F.2) |
where the last equality follows from an integration by parts. To justify this integration by parts, note that, for fixed , the -th component of , we have
for any fixed , , , since the asymptotic behavior of and are given by
as in Part 7 of Lemma E.1 and Part 11.A of Lemma E.3, respectively. Thus the last equality of (F.2) follows.
Appendix G Proof for Case I
By (F.3) and the decomposition
we have
where, for notational convenience and to control the size of expressions,
Further, by the triangle inequality and the fact , we have
where
| (G.1) | ||||
| (G.2) |
The proof of Case I will be completed by showing that each for is bounded by an integrable function. The theorem then follows by the dominated convergence theorem since since and in the expression of (F.1).
G.1
G.2
G.2.1 under Assumption A.2 with
By the Cauchy-Schwarz inequality, we have
| (G.5) |
Similarly, by the Cauchy-Schwarz inequality, we have
| (G.6) |
where the second inequality follows from the fact . Hence, by (G.5) and (G.6) with (G.2),
By the relationship
| (G.7) |
we have
| (G.8) |
where
| (G.9) |
is bounded by Part 1 of Lemma E.1. Further, by (G.8), we have
| (G.10) |
where given by (G.4) is bounded and
| (G.11) |
G.2.2 under Assumption A.2 with
Let
Note that and for any . Then, by Cauchy-Schwarz inequality, we have
| (G.12) |
where the second inequality with the constant follows from Lemma I.2, provided below in Appendix I. Similarly, by the Cauchy-Schwarz inequality, we have
| (G.13) |
where the second inequality with the constant follows from Lemma I.2, provided below in Appendix I, and from the fact . Hence, by (G.7), (G.12), (G.13) and with given by (G.9), we have
Therefore we have
| (G.14) | |||
where given by (G.4) is bounded and
The proof of Theorem 4.2 Case I is thus completed by applying the dominated convergence theorem to as noted above.
Appendix H Proof for case II
With , we have
| (H.1) |
By Lemma I.3 and the relationship (H.1), the integral included in (F.3) is rewritten as
where, again, with the notation
Similarly, by Lemma I.3 and the relationship (H.1), the integral included in (F.3) is rewritten as
Then given by (F.3) is rewritten as
By the triangle inequality and the fact ,
where
| (H.2) | ||||
| (H.3) |
For , as seen in Section G.1, we have . We will show integrability and integrability in Sub-sections H.1 and H.2, respectively.
H.1
Note the inequality
Then, in the first and second terms of (H.2), we have
where
and
Under Assumption A.2 on with , applying the same technique used in Sub-Section G.2.1, the integrability of
implies the integrability of . The integrability of is shown as follows;
| (H.4) | ||||
where
the inequality with follows from Part 2 of Lemma E.3 and the integrability of the right-hand side follows from Parts 3 and 5 of Lemma E.1.
H.2
Under Assumption A.2 on with , applying the same technique used in Sub-Section G.2.1, the integrability of
implies the integrability of . The integrability of is shown as follows;
where is given by (G.4), the first term is bounded by Parts 1 and 3 of Lemma E.1 and the second term is bounded by Part 8 of Lemma E.1.
Under Assumption A.2 on with , applying the same technique used in Sub-Section G.2.2 the integrability of
where implies the integrability of . The integrability of is shown as follows;
where is given by (G.4), the first term is bounded by Parts 1 and 2 of Lemma E.1 and the second term is bounded by Part 8 of Lemma E.1.
Appendix I Additional Lemmas used in Sections G and H
Let
| (I.1) | |||
Then we have a following result.
Lemma I.1.
Proof.
By Assumptions A.2 on ,
| (I.4) |
where
| (I.5) | |||
Let for the denominator and for the numerator of . By change of variables, the integral with respect to is rewritten as
where is defined by (I.3) and . Note
which are both positive and bounded from the above under and under Assumptions F.1–F.3 on and hence
under . Therefore we have
| (I.6) |
where
| (I.7) | |||
Note may be regarded as a non-central chi-square random variable with degrees of freedom and non-centrality parameter. For
we have
where the expected value is taken under the probability density given by
and
Since the correlation inequality gives
we have
For , we have
and hence
| (I.8) |
Finally, by (I.4), (I.5), (I.6), (I.7) and (I.8), we have
∎
Using Lemma I.1, we have the following result.
Lemma I.2.
Proof.
For Part 2, note the following relationship;
where the first inequality follows from the fact , the second inequality follows from Lemma I.1, the third inequality follows from Part 1 of Lemma E.2. The last inequality follows from Part 4 of Lemma E.2. Then, as in (I.12), the inequality (I.11) can be established. ∎
Proof.
First, note the following relationship;
By an integration by parts, the integral with respect to in the above is
where the first term becomes zero for any fixed under Assumptions. Then
which completes the proof. ∎
Appendix J Proof of Corollary 4.2, Part 3
Let where, as in (4.11),
for . Maruyama and Strawderman (2009) showed that
-
1.
is not monotonic,
-
2.
-
3.
in Part (iii) of Corollary 3.1, Part (iv) of Corollary 3.1 and Lemma 3.4, respectively. Kubokawa (2009) proposed a sufficient condition of to be minimax as follows;
where the upper bound is larger than the upper bound which Maruyama and Strawderman (2009) and Wells and Zhou (2008) applied. Note that is increasing in and that is decreasing in . Then the inequalities
as well as are a sufficient condition for minimaxity of . Let
For ,
which is nonnegative when
Hence Part 3 follows.
Appendix K Proof of Corollary 4.3
Let
where
Eventually set , and .
Note the underlying density is Gaussian and let
Note
Then we have
where
Also we have
Recall
Under the choice , we have
Let . Then the Bayes equivariant estimator is
| (K.1) |
When or equivalently as well as , this is a proper Bayes equivariant estimator. When or equivalently as well as , this is an admissible generalized Bayes equivariant estimator. Hence when and , the estimator (K.1) is admissible within the class of equivariant estimators.
Appendix L Proof of satisfaction of A.1–A.3 by (4.4)
Note the integrability of (L.1) under follows or as well as . Also the integrability of (L.1) under follows when . Further note Note also if as well as , and hence the prior on is proper.
Proof.
[Assumption A.2 with and ] When , by Tauberian theorem, we have, in (L.1) and (L.2),
| (L.3) |
which implies that
| (L.4) |
Recall and let . Then we have
by (L.3) and
by (L.4).
[Assumption A.2 with and ] When and , it follows that and . For this case take . Then , where and satisfies
[Assumption A.3] By Tauberian theorem, we have, in (L.1) and (L.2),
| (L.5) |
which implies that
| (L.6) |
Hence when , Assumption A.3A.3.1 is satisfied.
When , note
and
Then
and
Hence we get
which satisfies Assumption A.3(A.3.2)A.3.2.2. Hence Assumption A.3 is satisfied by . ∎
Appendix M Proof of Admissibility of for
In the Gaussian case, and , Kubokawa in his unpublished lecture note written in Japanese, showed that when , the estimator is admissible among all estimators. Here we generalize it for our general situation with the underlying density given by (1.1). For a general prior , we have
where and the last equality follows from an integration by parts. Hence the estimator is the generalized Bayes estimator with respect to any improper prior which does not depend on , say . Further let
where is given by (4.5). Clearly approaches as . Also for any fixed is integrable under the condition
| (M.1) |
since
Proof.
Let be the proper Bayes estimator with respect to . Then the Bayes risk difference of and with respect to is
Note
Then, by Cauchy-Schwarz inequality, we have
| (M.2) |
where and it is bounded under Assumptions F.1, F.2 and F.3F.3.1 on , as in Part 11.B of Lemma E.3. Further, by Lemma E.2 of Appendix E, for . Hence, by the dominated convergence theorem, we have as . By the Blyth sufficient condition, the admissibility of for follows. ∎
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